A fermionic approach to tunneling through junctions of multiple quantum wires Zheng Shi and Ian Affleck arXiv:1601.00510v2 [cond-mat.str-el] 2 Apr 2016 Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, Canada V6T 1Z1 (Dated: July 9, 2018) Junctions of multiple one-dimensional quantum wires of interacting electrons have received considerable theoretical attention as a basic constituent of quantum circuits. While results have been obtained on these models using bosonization and Density Matrix Renormalization Group (DMRG) methods, another powerful technique is based on direct perturbation theory in the bulk interactions, combined with the Renormalization Group (RG) and summed in the Random Phase Approximation (RPA). This technique has so far only been applied to the case where finite length interacting wires are attached to non-interacting Fermi liquid leads. We reformulate it in terms of the single-particle S-matrix, formally unifying treatments of junctions of different numbers of leads, and extend this method to cover the case of infinite length interacting leads obtaining results on 2-lead and 3-lead junctions in good agreement with previous bosonization and DMRG results. I. INTRODUCTION AND CONCLUSION A class of powerful theoretical approaches to junctions models the quantum wires as conformally invariant bulk Tomonaga-Luttinger liquids (TLL).1–6 In the spirit of boundary conformal field theory, at low energies the junction with its boundary operators should eventually renormalize to conformally invariant boundary conditions. Possible fixed points of the renormalization group (RG) flow are then postulated, and their various properties, such as zerotemperature conductance and operator scaling dimensions, are explored. Details of the RG flow, however, are largely open to conjecture except in the vicinity of these fixed points. These approaches are often consolidated with the technique of bosonization, as the elementary excitations of TLLs are bosonic in nature, and various boundary conditions imposed by the junction are often conveniently expressed in bosonic field variables. Alternate formalisms have been independently developed in the language of fermions. Consider first the junction system without bulk electron-electron interaction in the quantum wires; we further ignore local interactions so that this system becomes completely non-interacting. (For a free-fermion system, unless localized discrete states exist as is true for resonant tunneling, local interactions near the junction are irrelevant in the RG sense and do not affect leading order low-energy physics. We do not treat the case with discrete states in this paper.) A single-particle S-matrix determines the scattering basis, a new set of single-particle states which diagonalizes the non-interacting system. The bulk electron-electron interaction is then reintroduced and handled by perturbation theory, whose infrared divergence is resummed in an RG procedure. The S-matrix elements are now scale-dependent coupling constants. In the simplest approximation scheme, both the renormalized single-particle self-energy and the renormalized twoparticle vertex depend only on the renormalized S-matrix at every point in the RG flow; the S-matrix elements (or the transmission probabilities) are thus treated as the only running coupling constants of the theory, and by solving their RG equations we gain information about the conductance. This scheme was first adopted by Refs. 7, 8 to the first order in interaction; various generalizations include resonant tunneling with an energy-dependent S-matrix,9,10 second order perturbation theory in interaction,11,12 random phase approximation (RPA) in interaction,13–17 superconducting junctions,18,19 non-equilibrium transport,20 and refermionization of fixed points proposed by the bosonic theory,21 to list a few. In particular, it has been found that the RPA in the Tomonaga-Luttinger model with a linear dispersion reproduces various scaling dimensions of the conductance known from bosonic methods. (The term RPA has been used interchangeably with “ladder approximation” in Refs. 13–17.) An improved approximation, known as the functional RG method,22–24 explicitly studies the flows of single-particle self-energy and the two-particle vertex. Despite its basis on perturbation theory in interaction, the functional RG shows excellent agreement with analytic results for two-lead junctions at Luttinger parameter K = 1/2, and with numerical density matrix renormalization group (DMRG) data up to fairly strong interactions. The merit of these formalisms are that the crossover behavior between different fixed points can, in principle, be found to any order in interaction. Nevertheless, when the interaction becomes sufficiently strong in a junction of three wires (a “Y-junction”), the fixed points and the RG flow predicted by the RPA fermionic approach and the bosonic approach begin to differ qualitatively. Also a careful analysis reveals that, in the RPA approach, the β function of the S-matrix beyond one-loop order contains non-universal terms13,14,16 which depend on the precise cutoff scheme of the theory, and may potentially change its predictions. To our knowledge, many aspects of the junction problem have not been explored in the fermionic formalism. One such example is the well-known distinction between a semi-infinite TLL wire and a finite TLL wire connected to a Fermi liquid (FL) reservoir.1,25–36 There have been controversies on the nature of the conductance measured in a realistic experimental setup.34,36 However, if we consider the linear DC response to an externally applied bias voltage 2 FIG. 1. Sketch of the system with the number of leads N = 3. (rather than the total potential drop which includes induced polarization of the electronic density),31,32 then it has been predicted that the corresponding linear response coefficients (which we henceforth refer to as the “conductances”) with and without the FL reservoir (or “lead”) are generally different. (These coefficients are well-defined and can be studied numerically.)37 For instance, the conductance of a finite TLL wire attached to FL leads on both sides is e2 /h, irrespective of the interaction strength; on the other hand, the conductance of an infinite spinless TLL wire is Ke2 /h, where K is the Luttinger parameter. The Landauer formula based on a perfectly transmitting S-matrix alone cannot recover the Ke2 /h result. In existing literature employing the fermionic formalism, the case of FL leads has been well studied, but the effects of TLL leads on the conductance are not discussed. The reasons are twofold for our interest in the effects of TLL leads on the conductance from the fermionic perspective. At the fixed points well understood in the bosonic approach, such as the perfect transmission fixed point in the twolead junction and the chiral fixed points in the Y-junction, the agreement of these results in both approaches is a necessary validation of the fermionic approach. On the other hand, for the fixed points eluding the bosonic treatment, such as the maximally open fixed point of the Y-junction (known as the “M fixed point”),4,10,12,16,17,24 these results can be directly compared to numerics37 where available. In this work, we adopt the RPA fermionic approach to study the conductance tensor for a generic multi-lead junction in the presence of TLL leads. Our theory makes extensive use of the scattering basis transformation of the non-interacting part of the system; as a result it is explicitly formulated on the basis of the single-particle S-matrix (much like that in Ref. 8), and is formally independent of the number of wires. This stands in contrast to previous RPA treatments of junctions attached to FL leads, whose formulation is based on the renormalization of the conductance tensor instead and depends on the parametrization of the conductance tensor, different for two-lead junctions13–15 and Y-junctions.16,17 We derive a Landauer-type conductance formula, appropriate for the renormalized S-matrix, and recover the additional contribution from the TLL leads to the conductance, absent in the naive Landauer formalism. Our theory is applied to the two-lead junction and Y-junction problems, where in addition to verifying existing results on the fixed points and the phase diagrams, the conductance of the M fixed point attached to TLL leads is calculated. We summarize our findings below. The system consists of N quantum wires of interacting spinless electrons, numbered j = 1, 2, ..., N , meeting at a junction which we choose as the origin x = 0. We align the wires so that they are parallel to the +x axis; see Fig. 1. We assume that in the absence of interactions the junction is characterized by a single-particle S-matrix, Sjj ′ , independent of the energy of the incident/scattered electron. This is expected to be valid at low energies in the case of a non-resonant S-matrix, which we assume. We adopt a Tomonaga-Luttinger model for the electron-electron interaction in wire j, j Hint = ∞ 0 j † † dxg2 (x) ψjR (x) ψjR (x) ψjL (x) ψjL (x) , (1.1) j where ψjL/R are low-energy left- and right-moving electrons in wire j, and g2 (x → ∞) is a constant. A finite j j g2 (∞) = 0 corresponds to a TLL lead attached to wire j, while if g2 (∞) = 0 the junction is considered to be connected to an FL lead. We define a dimensionless interaction strength 3 j αj (x) = g2 (x) / (2πvF j ) , (1.2) where vF j is the Fermi velocity in wire j without interaction. The linear DC conductance tensor Gjj ′ of the junction is defined by Ij = j ′ Gjj ′ Vj ′ , where Ij is the current flowing away from the junction in wire j, and Vj ′ is the bias voltage applied on lead j ′ . In the absence of interactions, Gjj ′ is given by the Landauer formula GFL = jj ′ e2 δjj ′ − |Sjj ′ |2 . 2π (1.3) In the first order perturbation theory in αj , where the infrared singular corrections to physical observables are not resummed using RG methods, attaching a junction to TLL leads as compared to FL leads changes its linear DC conductance by GTLL − GFL = − jj ′ jj ′ e2 2π n 1 2 αn (∞) δjn − |Sjn | 2 δnj ′ − |Snj ′ | 2 . (1.4) In the first order RG-improved perturbation theory, the bare S-matrix will be replaced by the renormalized S-matrix at temperature T in Eqs. (1.3) and (1.4). In the RPA “bare” perturbation theory, the linear DC conductance tensors of the same junction attached to TLL leads and FL leads are related by GTLL = 1 − GFL G−1 c −1 GFL , (1.5) where 1 is the N × N identity matrix, and G−1 is the contact resistance tensor between the wires and leads, c G−1 jj ′ c = e2 2π −1 1 L 1 − Kj 2 −1 δjj ′ . (1.6) L Here the bulk Luttinger parameter of the lead j is given by Kj = (1 − αj (∞)) / (1 + αj (∞)). In the RPA RGimproved perturbation theory, the bare S-matrix is again replaced by the renormalized S-matrix at temperature T in Eqs. (1.3) and (1.5). 2 For a Z3 symmetric Y-junction at the maximally open M fixed point, |Sjj ′ | = 4/9 − δjj ′ /3. When the junction is attached to TLL leads with dimensionless interaction strength α and Luttinger parameter K L , at the first order the conductance tensor is GTLL,M = jj ′ 4 8 − α 9 27 e2 (3δjj ′ − 1) . 2π (1.7) In the RPA, the conductance at M becomes GTLL,M = jj ′ 4K L e2 (3δjj ′ − 1) . 3K L + 6 2π (1.8) Also, the temperature dependence of the conductance with LL leads, as dictated by Eq. (1.5), is governed by the same fixed point exponents as those for FL leads at temperatures above the inverse lengths of interacting wires. For instance, near the Z3 symmetric M fixed point, in the presence of a quadratic perturbation at the junction which preserves both Z3 symmetry and time-reversal symmetry, the correction to Eq. (1.7) [or Eq. (1.8)] scales as T −2α [or T −6(1−K)/(2+K) ]; on the other hand, for a quadratic perturbation at the junction which preserves Z3 symmetry but 2 breaks time-reversal symmetry, the correction to Eq. (1.7) [or Eq. (1.8)] scales as T α/3 [or T 3(1−K)(2−K)/(2+K) ].8,16,17 Whether a power law describes leading high-temperature or low-temperature behavior depends on the stability of M with respect to that perturbation. Eqs. (1.4)–(1.8) are the main results of this paper. 4 The rest of this paper is organized as follows. Section II elaborates on our model for a generic multi-lead junction, and calculates its linear DC conductance to the first order in interaction. Section III is based on perturbative RG, again to the first order in interaction. We derive the S-matrix RG equation in a Callan-Symanzik (CS) approach14 using the Kubo conductance calculated in Section II B. This establishes a modified Landauer formula involving the renormalized S-matrix in the case of FL leads. An additional contribution to the conductance, Eq. (1.4), is shown to arise from TLL leads. In Section IV, the conductance is found in the RPA to arbitrary order in interaction; we derive an S-matrix RG equation in the RPA, and again find the conductance [Eq. (1.5)]. Section V applies our results to the fixed points of 2-lead junctions and Y-junctions at the first order and in the RPA. In particular, we find the conductance at the M fixed point of a Z3 symmetric Y-junction attached to TLL leads, Eqs. (1.7) and (1.8). Open questions are discussed in Section VI. In Appendix A we show details of the conductance calculations up to the first order in interaction. The Wilsonian derivation of the RG equation for the S-matrix7 is reviewed in Appendix B. Finally, the RPA conductance calculations are explained in Appendix C. II. FIRST ORDER PERTURBATION THEORY OF KUBO CONDUCTANCE In this section, we establish the model Hamiltonian, and present our results for the linear DC conductance at the first order in interaction. A. Formulation of the problem The system is modeled by a Hamiltonian consisting of three parts: N j j H0,wire + Hint + H0,B . H= (2.1) j=1 j H0,wire is the non-interacting part of the Hamiltonian for wire j, quadratic in electron operators, while the quartic j Hint term of Eq. (1.1) describes the electron-electron interaction in wire j. The boundary term H0,B is quadratic, and is responsible for electron transfer between wires across the junction. For simplicity we assume that each wire only supports one single channel, and ignore quartic interactions between wires, at the junction and between the junction and the wires. In the continuum limit of the model, on each quantum wire we retain right- and left-movers in narrow bands of wave vectors around the Fermi points ±kF j : ψj (x) ≈ eikF j x ψjR (x) + e−ikF j x ψjL (x) = D −D dE i ψjR (E) e 2πvF j E vF j +kF j x + ψjL (E) e −i E vF j +kF j x , (2.2) where vF j is the Fermi velocity in wire j, the dispersion relation is E = Ej (k) = vF j k, and D ≪ vF j kF j is the high-energy cutoff. Left-movers ψjL are incident on the junction, scattered, and turned into right-movers ψj ′ R ; ψjL and ψj ′ R are not independent degrees of freedom, but related by the S-matrix of the junction [see also Eq. (2.5)]. The quadratic part of the wire Hamiltonian now reads j H0,wire ≈ ivF j ∞ 0 † † dx ψjL ∂x ψjL − ψjR ∂x ψjR (x) ≈ D −D † † dE E ψjR (E) ψjR (E) − ψjL (E) ψjL (E) , (2.3) To model the electron-electron interaction, we assume it is short-ranged and the system is away from half-filling, so that the Umklapp processes are unimportant. We further ignore processes where two chiral densities of the same † † † † chirality interact with one another, ψR ψR ψR ψR or ψL ψL ψL ψL ; these g4 processes38 renormalize the Fermi velocity but do not change the Luttinger parameter by themselves. For spinless fermions, this leaves us with only processes † † involving two chiral densities of different chiralities, or g2 processes, ψR ψR ψL ψL . The electron-electron interaction is then represented by a spatially variant g2 term as in Eq. (1.1). Along the lines of Ref. 7, viewing the electron-electron interaction as a perturbation, we can first diagonalize the quadratic part of the Hamiltonian. The resultant eigenstates, which form the so-called scattering basis, can be related to the S-matrix in the low-energy theory. Note that such a scattering basis transformation is independent of the 5 actual eigenstates of the fully interacting system; we are therefore always able to proceed with this transformation, regardless of whether the interaction is present at x → ∞. For non-resonant scattering, which we assume throughout this paper, the S-matrix elements Sjj ′ (E) ≡ Sjj ′ are independent of the electronic energy E, and the single-particle scattering state incident from wire j ′ with energy E ′ reads ∞ φ†′ (E ′ ) |0 = j 1 2πvF j dx j 0 ′ δjj ′ e −i vE x Fj ′ † ψjL (x) + Sjj ′ e i vE x Fj † ψjR (x) |0 + · · · , (2.4) where |0 corresponds to the filled Dirac sea, and the omitted terms represent contributions from the junction area. Inverting Eq. (2.4) we may express the original electrons ψ in terms of the scattering basis operators φ, N ∞ dE ′ ψjR (E) = dx 0 j ′ =1 D = j′ −D 1 i E x e vF j 2πvF j ∗ ′ 1 i E x Sjj ′ e vF j 2πvF j φj ′ (E ′ ) ′ −i dE Sjj ′ φj ′ (E ′ ) 2π E − E ′ − i0 (2.5a) Similarly D ψjL (E) = −D dE ′ i φj (E ′ ) 2π E − E ′ + i0 (2.5b) Now recast the Hamiltonian in the scattering basis. By definition, the quadratic part of the Hamiltonian is diagonal: N j H0,wire + H0,B = dE Eφ† (E) φj (E) j (2.6) j j=1 We insert the scattering basis transformation into the interaction Eq. (1.1). Allowing the energies to run freely from −∞ to ∞ and calculating the energy integrals using the method of residues,39 we find j Hint = ∞ 0 dE1 dE2 dE3 dE4 j dxg2 (x) (2π) l1 l2 l3 l4 2 2 vF j φ†1 (E1 ) φl2 (E2 ) φ†3 (E3 ) φl4 (E4 ) e l l i(−E1 +E2 +E3 −E4 ) v x Fj ∗ Sjl1 Sjl2 δjl3 δjl4 (2.7) This is a plausible manipulation, seeing that the scattering basis transformation should not introduce additional singularities at the band edge. Now j Hint = ∞ 0 dE1 dE2 dE3 dE4 j dxg2 (x) (2π) l1 l2 l3 l4 2 2 vF j ̺j1 l2 l3 l4 (E1 , E2 , E3 , E4 ; x) φ†1 (E1 ) φl2 (E2 ) φ†3 (E3 ) φl4 (E4 ) , l l l (2.8) where we introduce the function ̺j1 l2 l3 l4 (E1 , E2 , E3 , E4 ; x) ≡ l x 1 i(−E1 +E2 +E3 −E4 ) vF j ∗ i(−E3 +E4 +E1 −E2 ) v x F j S∗ S Sjl1 Sjl2 δjl3 δjl4 + e e jl3 jl4 δjl1 δjl2 . (2.9) 2 Note that we have symmetrized the function ̺ so that ̺j1 l2 l3 l4 (E1 , E2 , E3 , E4 ; x) = ̺j3 l4 l1 l2 (E3 , E4 , E1 , E2 ; x). This l l interaction is diagrammatically represented by the symmetric vertex in Fig. 2. We may well opt not to symmetrize ̺; however, the two created electrons E1 l1 and E3 l3 (or the two annihilated electrons E2 l2 and E4 l4 ) would be inequivalent in that case, and the diagrammatic bookkeeping would be more difficult. 6 FIG. 2. Diagrammatic representation of the electron-electron interaction. B. Kubo conductance We now compute the linear DC conductance in Kubo formalism. The current operator at coordinate x in wire j is first written in terms of the fermion fields: † † ˆ Ij (x) = evF j ψjR ψjR − ψjL ψjL (x) . (2.10) ˆ Note that Ij is not changed by the interaction; it is proportional to the commutator of the electron density with the Hamiltonian, but the interaction commutes with the electron density. Using Eq. (2.5) we find the imaginary time correlation function Ωjj ′ (x, x′ ; τ − τ ′ ) ≡ − Tτ Ij (x, τ ) Ij ′ (x′ , τ ′ ) to be Ωjj ′ (x, x′ ; τ − τ ′ ) =− e2 (2π) × e dǫ1 dǫ2 dǫ′ dǫ′ e 1 2 2 i ǫ′ −ǫ′ 2 1 v F j′ i ǫ2 −ǫ1 vF j ′ ′ j1 j2 j1 j2 x′ ∗ ′ Sj ′ j1 Sj ′ j2 ′ −e −i ǫ′ −ǫ′ 2 1 v F j′ x′ x ∗ Sjj1 Sjj2 − e ′ ′ δj ′ j1 δj ′ j2 −i ǫ2 −ǫ1 vF j x δjj1 δjj2 ′ Tτ φ†1 (ǫ1 , τ ) φj2 (ǫ2 , τ ) φ†′ (ǫ′ , τ ′ ) φj2 (ǫ′ , τ ′ ) 1 2 j j 1 H . (2.11) The imaginary time-ordered expectation value should be evaluated in the Heisenberg picture. The linear DC conductance Gjj ′ is then given by the retarded current-current correlation function Ω, Gjj ′ (x, x′ ) = lim lim ω→0 ηω →0+ i Ωjj ′ x, x′ ; ω + − Ωjj ′ (x, x′ ; 0) , ω (2.12) where again ω + ≡ ω + iηω . The coordinate dependence should vanish in the ω → 0 limit, since where exactly we apply the bias or measure the current is inconsequential in a DC experiment.4,30 Eq. (2.12) is now calculated in perturbation theory. Switching to the interaction picture, we perform a Wick decomposition of the time-ordered product, go to the frequency space and sum over the Matsubara frequencies. The retarded correlation function is then obtained by analytic continuation iωn → ω + ≡ ω + iηω where the ηω → 0+ limit is taken. The energy integrals are calculated afterwards, followed by real space integrals [which appear in Eq. (2.8)] in the end. Some details of this mostly standard calculation are given in Appendix A; here again we only show the final results. Feynman diagrams involved in the first order are shown in Fig. 3. In the absence of interaction, we have a single bubble diagram which leads to the usual linearized Landauer formula:40 (0) Gjj ′ = e2 δjj ′ − |Sjj ′ |2 2π (2.13) 7 FIG. 3. Diagrams contributing to the linear DC conductance at the first order in interaction. The second line shows the self-energy dressed bubble diagrams, while vertex correction diagrams are in the third line. Higher order diagrams can be classified into two basic types, namely self-energy diagrams and vertex corrections. At the first order, contributions from self-energy diagrams can be integrated into a Landauer-type formula: (1),SE (0) Gjj ′ + Gjj ′ = e2 2π δjj ′ − d(1) dǫ [−n′ (ǫ)] Sjj ′ (ǫ) F 2 (2.14) where the first order “dressed S-matrix” S d(1) is given by ∞ d(1) Sjj ′ (ǫ) = Sjj ′ − i dy αn (y) n 0 ∗ × Sjn Snn Snj ′ exp 2i (ǫ − ǫ′ ) dǫ′ nF (ǫ′ ) vF n y vF n + δjn Snn δj ′ n exp −2i (ǫ − ǫ′ ) y vF n ; (2.15) nF (ǫ) = 1/ eβǫ + 1 is the Fermi distribution at temperature β = 1/T , and αn (y) is defined in Eq. (1.2). For a d(1) non-interacting system Sjj ′ (ǫ) = Sjj ′ ; this is in agreement with our intuitive expectation. We now perform the y integral in a simple model. Let us assume that the junction is connected through wire n to a TLL or FL lead at x = Ln ; in other words, when x ≥ Ln , αn (x) = αn (∞) becomes a constant independent of x and dαn (x) /dx = 0. We further assume that the interaction inside the wire is also uniform, i.e. αj (x) = αj (0) + [αj (∞) − αj (0)] θ (x − Lj ) (2.16) where θ (x) is the Heaviside unit-step function. Integrating over y: d(1) Sjj ′ (ǫ) = Sjj ′ (ǫ) − dǫ′ n L nF (ǫ′ ) 2i ǫ−ǫ′ ) v n ∗ F n + α (0) Sjn Snn Snj ′ (αn (∞) − αn (0)) e ( n ′ − ǫ) 2 (ǫ −δjn Snn δj ′ n (αn (∞) − αn (0)) e L −2i(ǫ−ǫ′ ) v n Fn + αn (0) . (2.17) 8 The ǫ′ integral is infrared divergent, which prompts an RG resummation of leading logarithms. We will determine the renormalization of the S-matrix using Eq. (2.17) and discuss its implications in Section III. The vertex corrections, in the meanwhile, contribute a completely different type of terms: (1),VC Gjj ′ (x, x′ ) = − ∞ e2 i 2π dy αn (y) vF n 0 n 2 × lim lim + ω |Sjn | δj ′ n e iω + x vF j − vx ′ F j′ e ω→0 ηω →0 −δjn δj ′ n e −δjn δj ′ n e −iω + iω + x vF j x vF j + vx +v + 2 iω 2 + 2 −iω e x′ F j′ e x vF j − |Sjn | |Snj ′ | e x vF j +δjn |Snj ′ | e ′ F j′ 2iω + v y Fn −2iω + v y Fn + vx e x′ F j′ Fn θ x vF j θ y x − vF j vF n θ y θ − vF n θ y vF n y vF n − x′ vF j ′ − x′ vF j ′ y x′ − vF j ′ vF n ′ F j′ +v 2iω + v y e 2iω + v y Fn 2iω + v y Fn θ x y − vF n vF j (2.18) Integration by parts gives us 2iω + vF n yu dyαn (y) e 2iω + v y Fn = αn (yu ) e 2iω + vyu yl Fn − αn (yl ) e yl Fn 2iω + v yu − dy e 2iω + v y Fn yl dαn (y) , dy (2.19) where yu can be vF n x/vF j , vF n x′ /vF j ′ or ∞, and yl can be vF n x/vF j , vF n x′ /vF j ′ or 0. We can let x and x′ be sufficiently large so that yu > Ln is always satisfied; thus in the dαn /dy term in Eq. (2.19), yu may be replaced by Ln . If yu → ∞, the αn (yu ) term damps out due to the small imaginary part ηω , and Eq. (2.19) becomes in the ω → 0 and ηω → 0 limit 2iω + vF n yu dyαn (y) e 2iω + v y Ln Fn yl = −αn (yl ) − dy yl dαn (y) = −αn (Ln ) = −αn (∞) . dy (2.20a) On the other hand, if yu is finite, the αn (yu ) term will survive the ω → 0 and ηω → 0 limit: 2iω + vF n yu dyαn (y) e 2iω + v y yl Fn Ln = αn (yu ) − αn (yl ) − dy yl dαn (y) = αn (yu ) − αn (Ln ) = 0. dy (2.20b) Therefore, taking the DC limit explicitly in Eq. (2.18), we find wire n contributes to the vertex correction only when it is attached to a TLL lead, and the interaction inside the wire is immaterial: (1),VC Gjj ′ (x, x′ ) = − e2 2π n 1 2 αn (∞) δjn − |Sjn | 2 δnj ′ − |Snj ′ | 2 . (2.21) When αn (∞) = 0, as is the case for any wire n attached to an FL lead, the vertex correction due to n vanishes. III. FIRST-ORDER CALLAN-SYMANZIK PERTURBATIVE RG In this section, we analyze the result of Section II from the perspective of the CS formulation of RG,14 and present a modified Landauer formula involving the renormalized S-matrix in the case of FL leads, supplemented by vertex corrections from TLL leads. 9 Intuitively, once the renormalization flow of the S-matrix is stopped by a physical infrared cutoff, the renormalized Smatrix should represent the non-interacting part of the low-energy theory of the junction, and can be taken as an input to the Landauer formalism. However, such an argument does not address the role of the low-energy residual interaction, which turns out to be especially important in the case of TLL leads. Also, in principle, the Landauer formalism is well-founded only in the absence of inelastic scattering. We are therefore motivated to study the conductance in the CS formulation, which fully exposes possible deviations from the Landauer predictions. In the CS formulation of RG, we start from a field theory with a running cutoff D, and calculate low-energy physical observables (in our case the linear DC conductance tensor Gjj ′ ) as a function of the running coupling constants of the theory [in our case the S-matrix elements Sjj ′ (D)]. This is once again accomplished by perturbation theory in interaction, in formal analogy to Section II. However, the crucial difference is that we are now expressing certain lowenergy physical quantities in terms of running coupling constants, whereas in Section II we calculate the corresponding renormalized quantities in terms of bare coupling constants. We require that when D is greater than the energy scales at which the system is probed, namely the finite temperature T , Gjj ′ should be independent of D. Therefore, by allowing the cutoff to run from D to D − δD, where δD ≪ D, we can find the RG equation satisfied by the coupling constants Sjj ′ (D). Beginning from the simplest case where all leads are FL leads, αn (∞) = 0 for all n, the vertex correction Eq. (2.21) vanishes, and the full linear DC conductance to O (α) is given by Eq. (2.14). Reducing the cutoff from D to D − δD and demanding the right-hand side of Eq. (2.14) be a scaling invariant, we have δD D 2 d(1) ∗ d(1) −D dǫ [−n′ (ǫ)] F Sjj ′ (ǫ, D) d(1) dǫ [−n′ (ǫ)] Sjj ′ (ǫ, D) + F d(1) d(1) δSjj ′ (ǫ, D) + c.c. = 0 (3.1) where d(1) δSjj ′ (ω, D) ≡ Sjj ′ (ω, D) − Sjj ′ (ω, D − δD) , (3.2) d(1) Here Sjj ′ (ǫ, D) is Eq. (2.17) with the ǫ′ integral going from −D to D, and all S-matrix elements understood to be cutoff-dependent, Sjj ′ → Sjj ′ (D). Since the derivative of the Fermi function is peaked at the Fermi energy with width T , the δD integral in Eq. (3.1) approximately vanishes while D T ; Eq. (3.1) is thus automatically satisfied if Eq. (3.2) vanishes. The implication is that, at least in the case of FL leads, the renormalization of the conductance can be fully accounted for by the renormalization of the S-matrix. d(1) To the lowest order in δD, the condition that δSjj ′ (ω, D) = 0 is equivalent to δSjj ′ (ω, D) ≡ Sjj ′ (ω, D) − Sjj ′ (ω, D − δD) L nF (ǫ′ ) 2i ω−ǫ′ ) v n ∗ F n + α (0) = dǫ′ Sjn Snn Snj ′ (αn (∞) − αn (0)) e ( n 2 (ǫ′ − ω) δD n −δjn Snn δj ′ n (αn (∞) − αn (0)) e D L −2i(ω−ǫ′ ) v n Fn + αn (0) . (3.3) −(D−δD) stands for integration over fast modes. where δD = (D−δD) + −D If D |ω|, ǫ′ − ω can be approximated as ±2D, thus giving rise to a scaling contribution O (δD/D). If D vF n /Ln , exp (±i2DLn /vF n ) oscillates rapidly with D and is negligible; on the other hand, when D vF n /Ln , exp (±i2DLn/vF n ) ≈ 1. Finally, if D T , the factors nF (D) ≈ 0 and nF (−D) ≈ 1 are approximately independent of D. Therefore, to O (δD/D), Eq. (3.3) predicts that δSjj ′ = − independent of ω, provided D δD 2D n ∗ αn (D) Sjn Snn Snj ′ − αj (D) Sjj δjj ′ , (3.4) max {|ω| , T }. Here we have defined a cutoff-dependent interaction strength αn (D) ≡ αn (0) , D αn (∞) , D vF n /Ln . vF n /Ln (3.5) 10 FIG. 4. Dressing of the first order vertex correction diagrams by the first order self-energy diagrams. This means the renormalization will stop at the energy scale of the incident/scattered electron or the temperature, whichever is higher. In addition, the energy scale associated with the inverse length of wire n, vF n /Ln , determines whether the renormalization due to that wire is controlled by interaction strength in the wire αn (0) or that in the lead αn (∞): the effective interaction strength crosses over from αn (0) to αn (∞) as the D is reduced below vF n /Ln . We are now in a position to write down the RG equation for the S-matrix valid to O (α). Restoring the explicit cutoff dependence, we have − 1 dSjj ′ (D) =− d ln D 2 n ∗ αn (D) [Sjn (D) Snn (D) Snj ′ (D) − δj ′ n Snn (D) δnj ] (3.6) where the RG flow is cut off at the temperature T . This is the equation given in Refs. 7 and 8. It can be readily checked that Eq. (3.6) preserves the unitarity of the S-matrix. We pause to remark that, as the cutoff is reduced below the inverse length of one of the wires, renormalization due to that wire is governed only by the lead to which that wire is attached. This is reasonable because a junction of finite-length TLL wires attached to FL leads should, at low energies, renormalize into a junction connected directly to FL leads.4,14,30 Returning to the conductance analysis, once the cutoff D is reduced to the order of T , the perturbative correction d(1) to the S-matrix Sjj ′ (ǫ, D)− Sjj ′ (D) vanishes to the scaling accuracy; thus Sjj ′ (D = T ) may be used to approximate the dressed S-matrix in Eq. (2.14), and the conductance for a junction connected to FL leads is given by the modified Landauer formula, GFL = jj ′ e2 δjj ′ − |Sjj ′ (T )|2 , 2π (3.7) where the S-matrix is now fully renormalized according to Eq. (3.6), with the cutoff reduced to the temperature T . This is the Landauer-type formula invoked in Refs. 7, and 8. When some of the leads are TLL leads, corrections of Eq. (2.21) must also be taken into account. It is important to note, however, that in a CS analysis of the total conductance, Eq. (3.2) remains valid to O (α). This is because as the cutoff is lowered, Eq. (2.21) contributes additional terms of the form of α (∞) S ∗ δS to Eq. (3.1). However, by Eq. (3.2), δS is O (α); hence α (∞) S ∗ δS is O α2 , and is negligible to O (α). To calculate the total conductance at D = T with TLL leads, we go slightly beyond the first order and dress the O (α) vertex correction diagrams with O (α) self-energy diagrams, shown in Fig. 4. The bare S-matrix in Eq. (2.21) is then replaced by the dressed S-matrix, S d(1) : 11 FIG. 5. The RPA diagrammatics: (a) effective interaction in the RPA represented by thick wavy lines; (b) dressed propagator in the RPA, to O (δD/D) in RG, represented by thick straight lines; and (c) diagrams contributing to the Kubo conductance in the RPA. The dressed propagator in (b) is calculated to O (δD/D) only, because higher order terms in δD/D do not contribute to the renormalization of the S-matrix [Eq. (4.2)]— see Section III for an explanation in the first order context. (a) and (c) do not involve truncation at O (δD/D) because any renormalization of the interaction [Eq. (4.3)] and the conductance [Eqs. (4.1) and (4.5)] can be attributed to the renormalization of the S-matrix. Note that (c) features a thin interaction line (rather than a thick one) to avoid double-counting. d(1),VC Gjj ′ (x, x′ ) = − e2 2π n × δnj ′ − 1 αn (∞) δjn − 2 d(1) dǫ1 [−n′ (ǫ1 )] Sjn (ǫ1 ) F d(1) dǫ2 [−n′ (ǫ2 )] Snj ′ (ǫ2 ) F 2 2 . (3.8) d(1) This allows us to repeat our previous analysis for the case of FL leads, and further approximate Sjj ′ by Sjj ′ (D = T ). Thus the TLL leads contribute an additional conductance of GTLL − GFL = − jj ′ jj ′ e2 2π n αn (∞) 2 δjn − |Sjn (T )| 2 δnj ′ − |Snj ′ (T )| 2 . (3.9) Eqs. (3.6), (3.7) and (3.9) provide a comprehensive first-order picture for non-resonant tunneling through a junction: the interaction renormalizes the S-matrix, the renormalized S-matrix determines the conductance through a Landauertype formula if the junction is connected to FL leads, and the residual interaction further modifies the conductance if the junction is attached to TLL leads. As will be demonstrated in Section IV, this picture is by no means limited to the first order. IV. S-MATRIX RENORMALIZATION AND CONDUCTANCE IN THE RPA In this section, we extends our first-order RG analysis in Section III to arbitrary order in interaction under the RPA.13–17 The correlation function Eq. (2.11) is perturbatively evaluated for both self-energy diagrams and vertex corrections by the same procedures, except that the interaction is dressed with ring diagrams; see Fig. 5. We subsequently find the S-matrix RG equation in the CS scheme and express the conductance in terms of the renormalized S-matrix. This is once more a straightforward calculation, and we simply present the outcome, leaving the details for Appendix C. 2 Introduce the shorthand Wjj ′ (D) ≡ |Sjj ′ (D)| . The RPA self-energy diagrams give rise to a modified Landauer formula: GFL = jj ′ e2 [δjj ′ − Wjj ′ (T )] , 2π (4.1) 12 where the renormalization of the S-matrix is governed by a generalization of Eq. (3.6), − 1 dSjj ′ (D) =− d ln D 2n 1 n2 ∗ Sjn1 (D) Πn1 n2 (D) Sn2 n1 (D) Sn2 j ′ (D) − δj ′ n1 Π∗ 1 n2 (D) Sn2 n1 (D) δn2 j . n (4.2) The RPA-dressed interaction is Π (D) ≡ 2 [Q (D) − W (D)] −1 , (4.3) 1 + Kj (D) , 1 − Kj (D) (4.4) where Qjj ′ = Qj (D) δjj ′ , Qj (D) = with Kj (D) = (1 − αj (D)) / (1 + αj (D)) being the cutoff-dependent “Luttinger parameter” for wire j; αj (D) is given in Eq. (3.5). To lowest order in αj , Πij = δij αj . When all wires of the junction are attached to FL leads, in parallel with the O (α) calculation, Eq. (4.1) captures the entirety of the conductance. This is in agreement with the Kubo formula calculation in Refs. 13–17 in the language of chiral fermion densities. When some wires are attached to TLL leads, they again provide important corrections to the DC conductance. All RPA vertex correction diagrams dressed with RPA self-energy evaluate to GTLL − GFL = − jj ′ jj ′ e2 2π 1 [δjn1 − Wjn1 (T )] ΠL1 n2 [δn2 j ′ − Wn2 j ′ (T )] , n1 n2 2 n (4.5) where the residual effective interaction is ΠL = 2 QL − W (T ) −1 , (4.6) L and QL is given by Eq. (4.4) with Kj replaced by Kj = (1 − αj (∞)) / (1 + αj (∞)), the Luttinger parameter of the lead. Remarkably, if we define the DC contact resistance tensor between the wires and leads, G−1 c jj ′ = e2 2π −1 1 L 1 − Kj 2 −1 δjj ′ , (4.7) then Eq. (4.5) can be formally recast as GTLL = 1 − GFL G−1 c −1 GFL , (4.8) where 1 is the N × N identity matrix. The same relation has been derived in Refs. 4 and 6, which assume that the DC contact resistance between a finite TLL wire and an FL lead is not affected by the junction at the other end of the TLL wire. This intuitive assumption is reinforced by our calculations. We emphasize that the inclusion of the vertex correction diagrams does not change the RG equation of the S-matrix, Eq. (4.2). [The TLL leads do change the renormalization of the S-matrix through the scale-dependent interaction, Eq. (3.5).] The reason for this is as follows. Eq. (4.2) results from dressing the single particle propagator as shown in panel (b) of Fig. 5. The conductance is calculated in perturbation theory by replacing all bare single particle propagators (the thin lines) with the dressed ones (the thick lines) in the basic bubble diagram and the vertex correction diagrams; or equivalently, by replacing all bare S-matrix elements with the ones dressed with the RPA self-energy. As with the case at the first order, the RPA vertex correction diagrams do not introduce additional cutoff-sensitive integrals, and all cutoff-sensitive integrals originate from the dressed S-matrix. Therefore, the dressed S-matrix should be a cutoff-independent quantity when we apply the CS scheme to the conductance, regardless of whether the vertex correction diagrams contribute to the conductance. The form of Eq. (4.2) is thus independent of vertex corrections. 13 An immediate consequence of the robustness of the S-matrix renormalization is that, in the temperature range above the inverse lengths of the wires, the universal scaling exponents of the conductance versus temperature are the same for FL leads and TLL leads to the accuracy of the RG method. Since the temperature dependence of the residual interaction ultimately results from that of the renormalized W matrix, Taylor-expanding Eq. (4.5) in the vicinity of a fixed point, we find that the scaling exponents of the TLL lead conductance are none other than those of the W matrix, i.e. those of the FL lead conductance; in this temperature range the TLL leads merely modify the nonuniversal multiplicative coefficients to the power law. Therefore, for the temperature dependence of the conductance in Section I, we have directly quoted the FL lead results from Refs. 8, 16, and 17. We also note that Eq. (4.8) is always valid in the temperature range above the inverse lengths of the wires, which allows us to determine in the RPA the full temperature dependence of the TLL lead conductance using that of the FL lead conductance. The full crossover of FL lead conductance has been worked out analytically for two-lead junctions13–15 and for certain special cases of Y-junctions.16 Eqs. (4.1)–(4.8) are the central results of this paper. They show that at least in the RPA, in addition to the Landauer-type formula, TLL leads give rise to important corrections to the linear DC conductance which are also given in terms of the renormalized S-matrix. In the remainder of this paper, we implement these results in non-resonant tunneling through 2-lead junctions and Y-junctions. V. FIXED POINT CONDUCTANCE In this section we evaluate the conductance at several established fixed points of 2-lead junctions and Y-junctions attached to FL leads and TLL leads. The analysis is carried out at the first order in interaction [Eqs. (3.7) and (3.9)] and then in the RPA [Eqs. (4.1) and (4.8)]. In particular, we will examine the conductance of the maximally open M fixed point in the RPA for the Z3 symmetric Y-junction. For simplicity, the interactions are once more modeled by Eq. (2.16). We write αj (0), the interaction strength in wire j, simply as αj ; also when the junction is connected to TLL leads, we assume the interactions in wires and leads are uniform and identical, i.e. αj (∞) = αj . Of course, by definition αj (∞) = 0 for FL leads. 1. 2-lead junction In a 2-lead junction of spinless fermions away from resonance, solving the S-matrix RG equations [Eq. (3.6) at the first order and Eq. (4.2) in the RPA], we find that the only fixed points are the complete reflection fixed point [the N (Neumann) fixed point] and the perfect transmission fixed point [the D (Dirichlet) fixed point].7,13–15 At the N fixed point W12 = 0, the two wires are decoupled from each other, and we find the obvious result that the conductance GN ,FL = GN ,TLL = 0, irrespective of what leads the junction is attached to. jj ′ jj ′ On the other hand, at the D fixed point W12 = 1, the backscattering between the two wires vanishes. With FL leads GD,FL = e2 /2π (2δjj ′ − 1), as predicted by the naive Landauer formula; with TLL leads, Eq. (3.9) predicts jj ′ GD,TLL = jj ′ 1− α1 + α2 2 e2 (2δjj ′ − 1) 2π (5.1) at the first order, and Eq. (4.8) predicts GD,TLL = jj ′ 2K1 K2 e2 (2δjj ′ − 1) K1 + K2 2π (5.2) in the RPA. Here the RPA has recovered the famous result for the conductance of two semi-infinite TLL wires.29 2. Y-junction Even at the first order in interaction, the RG flow portrait for a Y-junction is more complicated than the two-lead junction.8 Solving Eq. (3.6), we find a “non-geometrical” M fixed point whose existence and transmission probabilities generally depend on the interaction strengths, in addition to the “geometrical” fixed points N , Aj and χ± . Provided the interactions are not too strong, these are also the only fixed points allowed in the RPA.17 N (complete reflection) and Aj (asymmetric) can be obtained by adding a third decoupled wire with label j to the N and D fixed points 14 of the two-lead junction respectively. The conductances at N and Aj are therefore a trivial generalization of the two-lead case, and we will focus on χ± and M alone. At the chiral fixed points χ± , in the absence of interaction, an electron incident from wire j is perfectly transmitted to wire j ± 1 (here we identify j + 3 ≡ j); thus the time-reversal symmetry is broken. The W matrix is given by Wjj ′ = (1 − δjj ∓ ǫjj ′ ) /2, where the anti-symmetric tensor ǫjj ′ is defined by ǫ12 = ǫ23 = ǫ31 = 1, ǫ21 = ǫ32 = ± ǫ13 = −1 and ǫjj = 0. At the first order, inserting the W matrix into Eqs. (3.7) and (3.9), we find Gχ ′ ,FL = jj e2 /2π (3δjj ′ − 1 ± ǫjj ′ ) /2, and ± ± Gχ ′ ,TLL − Gχ ′ ,FL = − jj jj e2 1 (αj + αj ′ ) 2π 2 3 − δjj ′ 2 + 1 (α1 + α2 + α3 ) (1 − δjj ′ ± ǫjj ′ ) . 2 (5.3) In the RPA, on the other hand, Eq. (4.8) gives the conductance at χ± with TLL leads as ± Gχ ′ ,TLL = 2 jj e2 Kj (K1 + K2 + K3 ) δjj ′ + (±K1 K2 K3 ǫjj ′ − Kj Kj ′ ) , 2π K1 + K2 + K3 + K1 K2 K3 (5.4) which agrees with the result of bosonization analysis.6 The presence of the M fixed point can be inferred in a Z3 symmetric time-reversal invariant Y-junction with attractive interactions: in this system, N is unstable, Aj is forbidden by Z3 symmetry, and χ± are forbidden by timereversal symmetry, so there must be at least one stable fixed point. The W matrix has generally interaction-dependent elements at M . At the first order, Wjj ′ = α1 α2 α3 /αj α1 α2 +α2 α3 +α3 α1 1− α1 α2 α3 /αj α1 α2 +α2 α3 +α3 α1 1− 2 , j = j′ α1 α2 α3 /αj′ α1 α2 +α2 α3 +α3 α1 . (5.5) , j = j′ We see explicitly that M obeys time-reversal symmetry, Wjj ′ = Wj ′ j . Demanding 0 ≤ Wjj ′ ≤ 1, we find that at the first-order M can only exist in the following situations: 1) α1 , α2 , α3 > 0; 2) α1 , α2 , α3 < 0; 3) α1 > 0, α2 > 0, α3 < −α1 , α3 < −α2 ; 4) α1 < 0, α2 < 0, α3 > −α1 , α3 > −α2 ; and situations equivalent to 3) and 4) up to permuted subscripts (e.g. (α1 , α2 , α3 ) → (α3 , α1 , α2 )). Substituting Eq. (5.5) into Eq. (3.9), we find that at the first-order the conductance at M obeys 2 GM,TLL jj ′ − GM,FL jj ′ = 2 e 2 − 2π (α1 +αα )(α2 +α3 )(α1 +α3 ) α2 (α1 + α2 + α3 − αj ) , j = j ′ j 2(α +α α +α α )3 1 2 2 e2 (α1 +α2 )(α2 +α3 )(α1 +α3 ) 2π 2(α1 α2 +α2 α3 +α3 α1 )3 3 3 1 αj αj ′ (α1 α2 + α2 α3 + α3 α1 ) − (α1 α2 α3 )2 αj αj′ , j = j′ . (5.6) Note that for Z3 symmetric interactions (αj = α), Wjj ′ = 1/9 + δjj ′ /3 becomes independent of the interaction strength. Now Wjj ′ produces the maximal transmission probability 8/9 allowed by unitarity in a Z3 symmetric S-matrix, and at the first order GM,TLL − GM,FL = − (8/27) α e2 /2π (2δjj ′ − 1). Compared to FL leads, TLL jj ′ jj ′ leads enhance conductance for attractive interactions and reduce conductance for repulsive interactions, as with the two-lead D fixed point. In the RPA, the W matrix of the M fixed point is generally cumbersome, but reduces to the aforementioned maximally transmitting W matrix for Z3 symmetric interactions. Eq. (4.8) then gives GM,TLL = jj ′ 4K e2 (3δjj ′ − 1) . 3K + 6 2π (5.7) This result supports the findings of Ref. 37. There the M fixed point conductance of a Y-junction of infinite TLL wires is computed numerically using DMRG, and conjectured to be Gjj ′ = e2 2Kγ (3δjj ′ − 1) , 2K + 3γ − 3Kγ 2π where it is suggested that the dimensionless parameter γ is 4/9 based on the non-interacting limit K = 1. (5.8) 15 VI. DISCUSSION AND OPEN QUESTIONS In this paper, using the fermionic RG formalism, we calculated the linear DC conductance tensor of a junction of multiple quantum wires. We showed, both at the first order and in the RPA, that a junction attached to FL leads has a conductance tensor which obeys a linearized Landauer-type formula with a renormalized S-matrix. TLL leads modify the conductance through vertex corrections, and the conductance with FL leads may be heuristically related to the conductance with TLL leads through the contact resistance between leads. In this section, we would like to discuss some of the questions left open in our approach. First, we have assumed that scattering by the junction is fully described by operators which are quadratic in conduction fermions and independent of other degrees of freedom. Local operators quartic in fermions are ignored, among others. This does not pose a threat to the first-order calculations, because any quartic local operator has a scaling dimension of at least 4 × 1/2 = 2 in the non-interacting case, and is necessarily highly irrelevant. However, it has been shown that sufficiently strong attractive bulk interactions can render quartic boundary operators relevant.4 † † An example is the electron pair hopping operator at the Z3 symmetric Y-junction, ψ1L ψ1R ψ3 ∂x ψ3 (x = 0) + h.c.: it is of dimension 3/K at the asymmetric fixed point A3 , where K is the Luttinger parameter of all three wires, and A3 sees wire 3 decoupled from perfectly connected wires 1 and 2. Apparently, for very strong interactions K > 3, this operator becomes relevant and can potentially dominate the physical properties of the stable fixed point. Unfortunately, the present RPA analysis does not predict a scaling exponent consistent with this operator;17 it is hence incomplete in this regard, and should not be carried too far into the regime of strongly attractive bulk interactions. A related issue is the existence of the D fixed points in the Y-junction. Predicted by the bosonic approaches4,6,41 but not the fermionic ones,16,17 these fixed points are only stable for strong attractive interactions. They are most notably characterized by Andreev reflections, even when electron-electron interaction is absent in the bulk. This hints at multi-particle scattering at the junction, and rules out the possibility to represent the D fixed points by singleparticle S-matrices. (Single-particle S-matrices with particle-hole channels are not feasible either since the D fixed points respect particle number conservation.)4 The D fixed points are not predicted by purely fermionic approaches, because the latter are based on the ansatz that the junction is always described by a single-particle S-matrix along the RG flow; but such an ansatz will likely be invalidated if, for instance, relevant quartic boundary operators are present. We are thus led to believe that the lack of D fixed points in the present RPA analysis does not refute their possible stability when the bulk interactions are strongly attractive. Indeed, the refermionization method adopted by Ref. 21 may be successfully used to describe the crossover from the “pair tunneling” D fixed point to the χ± fixed points in the vicinity of Luttinger parameter K = 3, with an S-matrix of free fermions which are not the original electrons. On the other hand, even when the bulk interactions are relatively weak, it is not a priori clear to what extent the RPA is successful. In the Tomonaga-Luttinger model (which we have adopted in our bulk quantum wires), the RPA is known to be exact due to the interaction which separately conserves the numbers of right- and left-movers.39 This is no longer the case once right- and left-movers become mixed up by the scattering at the junction. It has been pointed out that going beyond the RPA changes the renormalization of the S-matrix away from the “geometrical” fixed points, although all universal scaling exponents stay the same.13,14,16 As for the “non-geometrical” M fixed point in the Y-junction, its position is generally shifted when we go beyond the RPA. Remarkably, however, if the interaction is Z3 symmetric, not only the W matrix but also the scaling exponents at the M fixed point remain identical with the RPA results up to the third order in interaction.16 The agreement of our RPA result with the numerics of Ref. 37 is suggestive, but more work on vertex corrections is required to verify the validity of our RPA conductance at the Z3 symmetric M fixed point with TLL leads, Eq. (1.8). ACKNOWLEDGMENTS This work was supported in part by NSERC of Canada, Discovery Grant 36318-2009 (ZS and IA) and the Canadian Institute for Advanced Research (IA). The authors would like to acknowledge helpful discussions with D. Giuliano, L. I. Glazman, Y. Komijani and A. Rahmani during the course of this work, and also V. Meden and D. G. Polyakov for bringing multiple important references to their attention. The authors gratefully acknowledge the hospitality of GGI Florence where part of this work was done. Appendix A: Details of zeroth and first order perturbation theory In this appendix we present some of the crucial steps in the perturbative calculation of the conductance up to the first order in interaction, which lead to Eqs. (2.14) and (2.21). We go through the standard procedures for the 16 conductance calculation at the zeroth order in interaction, then highlight the treatments specific to the first order. 1. Zeroth order At the zeroth order, there is only one bubble diagram for the current-current correlation function. Wick’s theorem gives ′ Tτ φ†1 (ǫ1 , τ ) φj2 (ǫ2 , τ ) φ†′ (ǫ′ , τ ′ ) φj2 (ǫ′ , τ ′ ) 1 2 j j 1 ′ ′ = −δj2 j1 δ (ǫ2 − ǫ′ ) Gj2 (ǫ2 , τ − τ ′ ) δj1 j2 δ (ǫ1 − ǫ′ ) Gj1 (ǫ1 , τ ′ − τ ) . 1 2 (A1) Here G is the free scattering basis Matsubara Green’s function Gj (E, iωn ) = 1/ (iωn − E), ωn = (2n + 1) π/β. We have dropped the H subscript in Eq. (2.11) when switching to the interaction picture. Going to the frequency space, doing the standard Matsubara sum 1 β iωn Gj2 (E2 , iωn ) Gj1 (E1 , iωn − ipm ) = nF (E2 ) − nF (E1 ) , −ipm + E2 − E1 (A2) where pm = 2mπ/β is a bosonic frequency, and performing analytic continuation ipm → ω + ≡ ω + iηω (ηω → 0+ ) yield the zeroth order retarded correlation function, (0) Ωjj ′ x, x′ ; ω + = +δjj ′ e i(ǫ1 −ǫ2 ) x vF j e2 (2π)2 −v x′ F j′ dǫ1 dǫ2 j1 j2 − |Sj ′ j |2 e i(ǫ2 −ǫ1 ) nF (ǫ2 ) − nF (ǫ1 ) δjj ′ e ++ǫ −ǫ −ω 2 1 x vF j i(ǫ1 −ǫ2 ) + vx ′ F j′ − |Sjj ′ |2 e ′ x vF j − vx i(ǫ2 −ǫ1 ) x vF j F j′ + vx ′ F j′ . (A3) We have done the j1 and j2 sums using unitarity of the S-matrix. Employing contour techniques, we integrate over ǫ1 on (−∞, ∞) for the term proportional to nF (ǫ2 ), and integrate over ǫ2 on (−∞, ∞) for the term proportional to nF (ǫ1 ): (0) Ωjj ′ x, x′ ; ω + = − e2 2 (2π) e2 (2π)2 dǫ2 (2πi) nF (ǫ2 ) δ jj ′ e dǫ1 (2πi) nF (ǫ1 ) δjj ′ e iω + iω + x vF j − vx x vF j − vx ′ F j′ − 0 − |S jj ′ + 2 iω x vF j + vx iω + x vF j + vx | e ′ F j′ − 0 − |Sjj ′ |2 e ′ F j′ ′ F j′ . (A4) 2 We note that the |Sj ′ j | term vanishes because the associated singularities are on the wrong side of the contour. Now combine the nF (ǫ2 ) and nF (ǫ1 ) terms and restore the cutoff D, recalling that ǫ2 − ǫ1 = ω + . This gives (0) Ωjj ′ x, x′ ; ω + = i e2 2π dǫ2 nF (ǫ2 ) − nF ǫ2 − ω + δjj ′ e iω + x vF j − vx ′ F j′ + 2 iω − 0 − |Sjj ′ | e x vF j + vx ′ F j′ . (A5) Substituting into Eq. (2.12), taking the ηω → 0+ limit and then the ω → 0 limit, we obtain Eq. (2.13). 2. First order At the first order, as shown in Fig. 3, the bubble diagram is dressed by two types of self-energies: contraction of E1 with E2 or E3 with E4 in Eq. (2.8) (the “tadpole”), and contraction of E1 with E4 or E2 with E3 . In addition, there are two types of first order vertex correction diagrams, the “cracked egg” diagram and the ring diagram. 17 For the self-energy diagrams and the dressed conductance bubbles we need two more types of Matsubara frequency sums. The first one is 1 β iωn Gj (ǫ, iωn ) = nF (ǫ) . (A6) The second one is 1 β iωn ′ Gj1 (ǫ′ , iωn ) Gj1 (ǫ1 , iωn ) Gj2 (ǫ2 , iωn + ipm ) 1 1 d˜ ǫ 1 nF (˜) ǫ ′ (ǫ − ip ) − ǫ − (ǫ2 − ipm ) − ǫ1 2 2πi m 1 1 1 1 1 1 . × ′ ǫ + i0 − ǫ − ǫ − i0 − ǫ′ ǫ − i0 − ǫ ǫ + ipm − ǫ2 ǫ + i0 − ǫ1 ˜ ˜ ˜ ˜ 1 1 1 ˜ = nF (ǫ2 ) (A7) To compute this sum, we consider the following contour integral, dz 1 1 1 , nF (z) 2πi z − ǫ′ z − ǫ1 z + ipm − ǫ2 1 (A8) where the integration contour is wrapped around the branch cut on the real axis,42 so that poles inside the contour are z = iωn (n running over all integers) and also z = ǫ2 − ipm . The nF (ǫ2 ) term in Eq. (A7) comes from z = ǫ2 − ipm , and the nF (˜) term comes from the branch cut z = 0. ǫ We ignore the tadpole-type self-energy diagrams, again on the grounds that they only modify the chemical potential. The other type of self-energy diagrams turn out to dress the S-matrix as in Eq. (2.17). One instance of these diagrams reads (1),SE,non-tadpole,1 Ωjj ′ × e i (ǫ2 −ǫ1 ) x vF j (x, x′ ; τ − τ ′ ) = − ∗ Sjj1 Sjj2 − e β × (−) ′ × δj2 j1 δ (ǫ2 − (ǫ2 −ǫ1 ) x vF j n ′ ǫ1 ) Gj2 (ǫ2 , τ − dǫ1 dǫ2 dǫ′ dǫ′ 1 2 2 (2π) ′ ′ j1 j2 j1 j2 δjj1 δjj2 e i (ǫ′ −ǫ′ ) 2 1 v F j′ x′ dE1 dE2 dE3 dE4 n dy g2 (y) dτ1 0 −i e2 2 l1 l2 l3 l4 ′ τ ′ ) δj2 l1 δ (ǫ′ 2 2 (2π) vF n ∗ ′ Sj ′ j1 Sj ′ j2 − e ′ −i (ǫ′ −ǫ′ ) 2 1 v F j′ x′ ′ ′ δj ′ j1 δj ′ j2 ̺n l2 l3 l4 (E1 , E2 , E3 , E4 ; y) l1 ′ − E1 ) Gj2 (ǫ′ , τ ′ − τ1 ) 2 × δl2 l3 δ (E2 − E3 ) Gl2 (E2 , 0) δl4 j1 δ (E4 − ǫ1 ) Gj1 (ǫ1 , τ1 − τ ) . (A9) Going to the frequency space, performing Matsubara sums and analytic continuation, we find (1),SE,non-tadpole,1 Ωjj ′ × e i (ǫ2 −ǫ1 ) x vF j × (−) x, x′ ; ω + = − ∗ Sjj1 Sjj2 − e n dy g2 (y) n −i e2 (ǫ2 −ǫ1 ) x vF j dE2 2 2 (2π) vF n dǫ1 dǫ2 dǫ′ 2 2 (2π) ′ j1 j2 j2 δjj1 δjj2 e i (ǫ′ −ǫ2 ) 2 v F j′ x′ ∗ ′ Sj ′ j2 Sj ′ j2 − e −i (ǫ′ −ǫ2 ) 2 v F j′ x′ ′ δj ′ j2 δj ′ j2 nF (E2 ) ̺n2 nnj1 (ǫ′ , E2 , E2 , ǫ1 ; y) 2 j′ 1 d˜ ǫ 1 1 − nF (˜) ǫ (ǫ2 − ω + ) − ǫ′ (ǫ2 − ω + ) − ǫ1 2πi ǫ + ω + − ǫ2 ˜ 2 1 1 1 1 . − ǫ + i0 − ǫ′ ǫ + i0 − ǫ1 ˜ ˜ ǫ − i0 − ǫ′ ǫ − i0 − ǫ1 ˜ ˜ 2 2 × nF (ǫ2 ) × (A10) 18 Carrying out the ǫ1 and ǫ′ integrations, and also the ǫ2 integration in the nF (˜) term, this becomes in the x, x′ → ∞ ǫ 2 limit (1),SE,non-tadpole,1 x, x′ ; ω + = − Ωjj ′ × (−) e2 2π dǫ2 nF (ǫ2 ) − nF ǫ2 − ω + 1 iω+ dE2 nF (E2 ) e vF n 2 dy αn (y) n x vF j + vx ′ F j′ −2i(E2 −(ǫ2 −ω + )) v y 2i E − ǫ −ω + )) v y ∗ ∗ ∗ Fn + δ Fn . × Sjj ′ Sjn Snn Snj ′ e ( 2 ( 2 jn Snn δj ′ n e (A11) This is just one of the four terms which reproduce Eqs. (2.14) and (2.15) when inserted in Eq. (2.12). Another identical term comes from contracting E1 with E4 (completely equivalent to contracting E2 with E3 which we have done). The remaining two terms have all their electron propagators reverted, so that their contributions to the conductance are the complex conjugate of the first two terms. This concludes the derivation of Eqs. (2.14) and (2.15). Neither type of vertex corrections to the conductance requires Matsubara sums other than Eq. (A2). An example of the ring diagram is (1),VC,ring,1 Ωjj ′ × e i (ǫ2 −ǫ1 ) x vF j ∗ Sjj1 Sjj2 β × (−) e2 (x, x′ ; τ − τ ′ ) = − −e −i (2π)2 (ǫ2 −ǫ1 ) x vF j dǫ1 dǫ2 dǫ′ dǫ′ 1 2 ′ ′ j1 j2 j1 j2 0 n (ǫ′ −ǫ′ ) 2 1 v F j′ x′ dE1 dE2 dE3 dE4 n dy g2 (y) dτ1 e δjj1 δjj2 i 2 2 (2π) vF n l1 l2 l3 l4 ∗ ′ Sj ′ j1 Sj ′ j2 ′ −e −i (ǫ′ −ǫ′ ) 2 1 v F j′ x′ ′ ′ δj ′ j1 δj ′ j2 ̺n l2 l3 l4 (E1 , E2 , E3 , E4 ; y) l1 ′ ′ × δj2 l1 δ (ǫ2 − E1 ) Gj2 (ǫ2 , τ − τ1 ) δl4 j1 δ (ǫ′ − E4 ) Gj1 (ǫ′ , τ1 − τ ′ ) 1 1 ′ ′ × δj2 l3 δ (ǫ′ − E3 ) Gj2 (ǫ′ , τ ′ − τ1 ) δj1 l2 δ (E2 − ǫ1 ) Gj1 (ǫ1 , τ1 − τ ) . 2 2 (A12) Going to the frequency space, performing Matsubara sums and analytic continuation: (1),VC,ring,1 x, x′ ; ω + = − Ωjj ′ × e × i (ǫ2 −ǫ1 ) x vF j ∗ Sjj1 Sjj2 − e n dy g2 (y) n −i e2 (2π)2 dǫ1 dǫ2 dǫ′ dǫ′ 1 2 ′ ′ j1 j2 j1 j2 (ǫ2 −ǫ1 ) x vF j 1 2 (2π)2 vF n δjj1 δjj2 e i (ǫ′ −ǫ′ ) 2 1 v F j′ ̺n2 j1 j2 j1 (ǫ2 , ǫ1 , ǫ′ , ǫ′ ; y) ′ ′ 2 1 j x′ ∗ ′ Sj ′ j1 Sj ′ j2 − e ′ −i (ǫ′ −ǫ′ ) 2 1 v F j′ x′ ′ ′ δj ′ j1 δj ′ j2 nF (ǫ2 ) − nF (ǫ1 ) nF (ǫ′ ) − nF (ǫ′ ) 2 1 . ǫ2 − ǫ1 − ω + ǫ′ − ǫ′ + ω + 2 1 (A13) Integrating over the energies as before, we find (1),VC,ring,1 Ωjj ′ × x, x′ ; ω + = − dy αn (y1 ) n −δjn δj ′ n e −δjn δj ′ n e −iω + iω + x vF j x vF j − |Sjn |2 |Snj ′ |2 e e2 2π dǫ1 dǫ′ (−) nF ǫ1 + ω + − nF (ǫ1 ) nF ǫ′ − ω + − nF (ǫ′ ) 1 1 1 iω + 1 1 2 |Sjn | δj ′ n e vF n 2 + vx +v iω + ′ F j′ e x′ F j′ x vF j e 2iω + v y Fn −2iω + v y Fn + vx θ θ x vF j − vx ′ F j′ e x vF j θ y x − vF j vF n 2iω + v y θ y vF n − ′ F j′ e 2iω + v y Fn + δjn |Snj ′ |2 e Fn y vF n − θ y vF n − x′ vF j ′ x′ vF j ′ y x′ − ′ vF j vF n −iω + x vF j − vx ′ F j′ e 2iω + v y Fn θ x y − vF n vF j . (A14) 19 There exists an analogous term with all electron lines reverted. Upon substitution into Eq. (2.12) these two terms reproduce Eq. (2.18). Finally, by summing over all dummy wire indices, we can show that the “cracked egg” contribution to the DC conductance is proportional to δjj ′ . On the other hand, due to current conservation and the absence of equilibrium currents, the full DC conductance Gjj ′ obeys Gjj ′ = Gjj ′ = 0; (A15) j′ j this must also be true at O (α). Since Eq. (A15) is already satisfied by Eqs. (2.14) and (2.21), it must also be separately satisfied by the “cracked egg” diagrams. But j δjj ′ = j ′ δjj ′ = 1, and we infer that the “cracked egg” diagrams must be identically zero. Appendix B: Details of the Wilsonian approach to S-matrix renormalization In this appendix, we review the derivation of the S-matrix RG equation using the Wilsonian scaling approach in Ref. 7. Starting from Eq. (2.8), we reduce the energy cutoff D to D − δD (δD ≪ D), and integrate out the so-called “fast modes” with energies in one of the two slices (−D + δD, −D) and (D − δD, D). This procedure generates corrections of O (αδD/D) to the quadratic part of the Hamiltonian [Eq. (2.6)] as well as the quartic part [Eq. (2.8)]. We assume that the corrections to the quartic part are unimportant; the rationale is that the quartic part originates entirely from the bulk, so it should renormalize independently of the junction. In fact, since the quartic part is free of Umklapp processes, it should be exactly marginal in the RG sense.43 Meanwhile, the renormalized quadratic part becomes off-diagonal and must be diagonalized with a new scattering basis, which is in turn associated with a running (i.e. cutoff-dependent) S-matrix. The quadratic correction generated by Eq. (2.8) reads j δH0 = −2 ∞ 0 D j dxg2 (x) dE1 δD l1 l2 l3 −D dE2 dE3 2 (2π)2 vF j ̺j1 l2 l3 l1 (E1 , E2 , E3 , E1 ; x) nF (E1 ) φ†3 (E3 ) φl2 (E2 ) . l l (B1) The E2 E3 contraction is equivalent to the E1 E4 contraction; hence the factor of 2. The E1 E2 and E3 E4 contractions 2 are discarded because, once we sum over l taking into account the S-matrix unitarity l1 |Sjl1 | = 1, we find they 43 only harmlessly shift the chemical potential. We let φ′ be the renormalized scattering basis after integrating out fast modes. φ′ is related to φ by another δ S-matrix, Sjj ′ , which only weakly deviates from the N × N identity matrix:  i −i dE ′  φ′ (E ′ ) + ′ + i0 j 2π E − E E − E ′ − i0 φj (E) = The inverse transformation is obtained by calculating anti-commutators: φ′ ′ (E ′ ) j † = j j′  δ Sjj ′ (E ′ ; E) φ′ ′ (E ′ ) j dE i −i δ ′+ S δ ′ (E ′ ; E) φ† (E) j ′ + i0 jj 2π E − E E − E ′ − i0 jj (B2) (B3) By definition φ′ diagonalizes the renormalized quadratic Hamiltonian,   j H0,wire + H0,B + j j  † j δH0 , φ′ ′ (E ′ )  = E ′ φ′ ′ (E ′ ) j j Substituting Eq. (B3) into the above, we find to O (δD/D) † (B4) 20 ∞ ′ i 2D−E−E αj ′ (x) αj (x) ∗ −i 2D−E−E x ∗ vF j + Sjj Sjj ′ e S j ′ j ′ S j ′ j e vF j ′ vF j vF j ′ 0 2D+E+E ′ 2D+E+E ′ −i v ′ x αj ′ (x) αj (x) ∗ i x ∗ Fj S Sjj ′ e vF j Sj ′ j ′ Sj ′ j e + +nF (−D) vF j jj vF j ′ δ i δjj ′ − Sjj ′ (E ′ ; E) = δD dx nF (D) ′ x (B5) For the simple model Eq. (2.16), integrating over x, we find   ′  −i 2D−E−E Lj vF j − αj (0) (αj (0) − αj (∞)) e δ ∗ δjj ′ − Sjj ′ (E ′ ; E) = δD nF (D) Sjj Sjj ′  2D − E − E ′   ′ ′ i 2D−E−E Lj′ i 2D+E+E Lj vF j − αj (0) − αj ′ (0)  (αj ′ (0) − αj ′ (∞)) e vF j′ ∗ ∗ Sjj Sjj ′ (αj (0) − αj (∞)) e +Sj ′ j ′ Sj ′ j + nF (−D) ′ ′ 2D − E − E −2D − E − E  ′ −i 2D+E+E Lj′ v F j′ − αj ′ (0)  ∗ (αj ′ (0) − αj ′ (∞)) e (B6) +Sj ′ j ′ Sj ′ j  −2D − E − E ′ When we assume D max {|E| , |E ′ | , T }, and apply the same considerations below Eq. (3.3), Eq. (B6) becomes δ Sjj ′ (E; E ′ ) = δjj ′ − δD ∗ ∗ αj (D) Sjj Sjj ′ − αj ′ (D) Sj ′ j ′ Sj ′ j . 2D (B7) The renormalized S-matrix S + δS relates φ′ to the original fermions ψ. Inserting Eq. (B2) into Eq. (2.5) we find that δS and S δ obey the simple matrix relation δS = SS δ − S, and according to Eq. (B7), δSjj ′ is given by none other than Eq. (3.4). Thus to the first order in interaction the CS approach and the Wilsonian approach predict the same S-matrix renormalization, Eq. (3.6). Appendix C: Details of the RPA In this appendix we expound the RPA calculations that lead to Eqs. (4.2) and (4.5). 1. RPA conductance The RPA self-energy beyond the first order involves a new type of Matsubara sum. For instance, at the third order in interaction, we need 1 β ipm 1 nF (E4 ) − nF (E3 ) nF (E8 ) − nF (E7 ) ipm + E4 − E3 ipm + E8 − E7 i (pm + ωn ) − E2 nF (E4 ) − nF (E3 ) nF (E8 ) − nF (E7 ) E2 − iωn + E4 − E3 E2 − iωn + E8 − E7 d˜ ǫ 1 nF (E4 ) − nF (E3 ) nF (E8 ) − nF (E7 ) + nB (˜) ǫ 2πi ǫ + iωn − E2 (˜ + i0) + E4 − E3 (˜ + i0) + E8 − E7 ˜ ǫ ǫ nF (E4 ) − nF (E3 ) nF (E8 ) − nF (E7 ) − (˜ − i0) + E4 − E3 (˜ − i0) + E8 − E7 ǫ ǫ = nF (E2 ) (C1) where nB (ǫ) = 1/ eβǫ − 1 is the Bose distribution. To derive Eq. (C1) we again wrap the integration contour around the branch cut at the real axis. The fraction with numerator nF (E3 ) − nF (E4 ) originates from the fermion loop with loop energy E3 and E4 ; at the lth order there will be l − 1 loops present. ipm is the bosonic frequency carried by the interaction lines; after ipm , iωn is also summed over following Eq. (A7). 21 After we perform analytic continuation and integrate over the loop momenta, as x, x′ → ∞, the three most important terms in the correlation function at the third order are (3),SE,RPA,E2 x, x′ ; ω + = Ωjj ′ e2 iω+ e 2π x vF j + vx ′ F j′ dǫ′ nF (ǫ′ ) − nF ǫ′ − ω + 1 1 1 ∞ × ˜ ˜ ˜ d˜1 d˜2 d˜3 αn1 (vF n1 y1 ) αn2 (vF n2 y2 ) αn3 (vF n3 y3 ) y y y n1 n2 n3 0 2 2 Sjj ′ ˜2 dE2 nF (E2 ) E2 ˜ y y y ∗ ∗ × Sjn3 Sn1 n3 Sn1 j ′ |Sn2 n1 | |Sn3 n2 | e2iE2 (˜1 +˜2 +˜3 ) ˜ y y y ∗ ∗ y ˜ y ˜ +Sjn3 Sn1 n3 Sn1 j ′ δn3 n2 δn2 n1 θ (˜1 − y2 ) θ (˜3 − y2 ) e2iE2 (˜1 −˜2 +˜3 ) ˜ y y y ∗ y ˜ +δjn3 δn3 n1 Sn1 j ′ δn3 n2 |Sn2 n1 |2 θ (˜2 − y3 ) e2iE2 (˜1 +˜2 −˜3 ) ˜ y y y ∗ y ˜ +Sjn3 δn3 n1 δn1 j ′ |Sn3 n2 |2 δn2 n1 θ (˜2 − y1 ) e2iE2 (−˜1 +˜2 +˜3 ) ˜ y y y ∗ y ˜ y ˜ +δjn3 Sn3 n1 δn1 j ′ δn3 n2 δn2 n1 θ (˜2 − y1 ) θ (˜2 − y3 ) e2iE2 (−˜1 +˜2 −˜3 ) (C2a) ˜ where we have substituted yn = yn /vF n and E2 = E2 − ǫ′ + ω + , ˜ 1 (3),SE,RPA,˜+ ǫ Ωjj ′ ′ x, x ; ω e2 iω+ = e 2π + x vF j + vx ′ F j′ dǫ′ nF (ǫ′ ) − nF ǫ′ − ω + 1 1 1 Sjj ′ ∞ × ˜ ˜ ˜ d˜1 d˜2 d˜3 αn1 (vF n1 y1 ) αn2 (vF n2 y2 ) αn3 (vF n3 y3 ) y y y n1 n2 n3 d˜nB (˜) ǫ2 ǫ ǫ ˜ 0 2 ǫ y y y ∗ y ˜ y ˜ × δjn3 δn3 n1 Sn1 j ′ δn3 n2 |Sn2 n1 | θ (˜2 − y3 ) θ (˜3 − y1 ) e2i(˜+i0)(˜1 +˜2 −˜3 ) 2 ǫ y y y ∗ y ˜ y ˜ +Sjn3 δn3 n1 δn1 j ′ |Sn3 n2 | δn2 n1 θ (˜2 − y1 ) θ (˜1 − y3 ) e2i(˜+i0)(−˜1 +˜2 +˜3 ) ǫ y y y ∗ y ˜ y ˜ +δjn3 Sn3 n1 δn1 j ′ δn3 n2 δn2 n1 θ (˜2 − y1 ) θ (˜2 − y3 ) e2i(˜+i0)(−˜1 +˜2 −˜3 ) (C2b) and finally (3),SE,RPA,˜− ǫ x, x′ ; ω + = Ωjj ′ e2 iω+ e 2π x vF j + vx ′ F j′ dǫ′ nF (ǫ′ ) − nF ǫ′ − ω + 1 1 1 Sjj ′ ∞ × ˜ ˜ ˜ d˜1 d˜2 d˜3 αn1 (vF n1 y1 ) αn2 (vF n2 y2 ) αn3 (vF n3 y3 ) y y y n1 n2 n3 d˜nB (˜) ǫ2 ǫ ǫ ˜ 0 ǫ y y y ∗ y ˜ y ˜ × −δjn3 δn3 n1 Sn1 j ′ δn1 n2 |Sn2 n3 |2 θ (˜2 − y1 ) θ (˜3 − y1 ) e2i(˜−i0)(˜1 −˜2 −˜3 ) ǫ y y y ∗ y ˜ y ˜ −δjn3 Sn3 n1 δn1 j ′ δn1 n2 δn2 n3 θ (˜1 − y2 ) θ (˜3 − y2 ) e2i(˜−i0)(−˜1 +˜2 −˜3 ) 2 ǫ y y y ∗ y ˜ y ˜ −Sjn3 δn3 n1 δn1 j ′ |Sn1 n2 | δn2 n3 θ (˜2 − y3 ) θ (˜1 − y3 ) e2i(˜−i0)(−˜1 −˜2 +˜3 ) 2 2 ǫ y y y ∗ −δjn3 Sn3 n1 δn1 j ′ |Sn1 n2 | |Sn2 n3 | e2i(˜−i0)(−˜1 −˜2 −˜3 ) (C2c) plus similar terms with all electron lines reverted. yj ≡ yj /vF j runs between 0 and ∞, j = 1, 2, 3. These three terms ˜ come from lines 2, 3 and 4 of Eq. (C1) respectively. In the DC limit, the zeroth order contribution and the self-energy corrections to the conductance again constitute a Landauer-type formula with a dressed S-matrix, similar to Eq. (2.14). Now we reduce the cutoff and demand the conductance be cutoff-independent. Once the y integrals are performed, it is obvious that the cutoff-sensitive integrals ˜ are the E2 integral and the ˜ integral. ǫ We are in a position to discuss the real space integrals. We first focus on the simplest case where the interactions in wires and leads are uniform and identical, αn1 (y) = αn1 for any n1 , so that all α’s factor out. At the third order, we find the following integrals: ∞ I1 E + ≡ d˜1 d˜2 d˜3 eiE y y y 0 + (˜1 +˜3 ) iE + (˜1 −2˜2 +˜3 ) y y y y y e θ (˜1 − y2 ) θ (˜3 − y2 ) y ˜ y ˜ (C3) 22 which appears alongside the factors δn1 n2 δn2 n3 , and ∞ d˜1 e2iE y + y1 ˜ = 0 i 2E + (C4) which appears alongside, for example, Wn1 n2 Wn2 n3 . [More accurately, Eq. (C4) comes with each “node” n2 as long ˜ as n2 is not sandwiched between two Kronecker δ factors.] Here E + ≡ E + i0 may be replaced by E2 or (±˜ + i0). ǫ At higher orders, we need to evaluate the integral 2M+1 IM E + ≡ ∞ d˜l eiE y + M 0 l=1 eiE (˜1 +˜2M +1 ) y y + (˜2j−1 +˜2j+1 −2˜2j ) y y y j=1 θ (˜2j−1 − y2j ) θ (˜2j+1 − y2j ) y ˜ y ˜ (C5) This is accompanied by a string of 2M consecutive δ factors uninterrupted by W factors, δn1 n2 δn2 n3 · · · δn2M n2M +1 . We will prove in Section C 2 that IM E + = i 2E + 2M+1 CM (C6) where CM = (2M )!/ [M ! (M + 1)!] is the M th Catalan number.44 The first few Catalan numbers 0 ≤ M ≤ 5 are 1, 1, 2, 5, 14, 42. −1 ˜ At this stage we can combine the (E + ) factors in Eqs. (C4) and (C6) with the E2 or ǫ factors. At each order ˜ −1 + there will be a single (E ) factor left unpaired, which gives the leading-log renormalization δD/D as the cutoff is reduced from D to D − δD. Collecting terms of all orders we see the S-matrix RG equation is of the form of Eq. (4.2), but the interaction Π (D) is given by αn1 αj αj ′ αj Πjj ′ Wjj ′ + = δjj ′ + [δjn1 δn1 j ′ + Wjn1 Wn1 j ′ ] 2 2 2 2 2 n1 αn1 αn2 [δjn1 δn1 n2 Wn2 j ′ + Wjn1 δn1 n2 δn2 j ′ + Wjn1 Wn1 n2 Wn2 j ′ ] + 2 2 n1 n2 αn1 αn2 αn3 + [2δjn1 δn1 n2 δn2 n3 δn3 j ′ + δjn1 δn1 n2 Wn2 n3 Wn3 j ′ 2 2 2 n n n 1 2 3 +Wjn1 δn1 n2 δn2 n3 Wn3 j ′ + Wjn1 Wn1 n2 δn2 n3 δn3 j ′ + Wjn1 Wn1 n2 Wn2 n3 Wn3 j ′ ] + · · · (C7) The rules to write down terms in Eq. (C7) are as follows. At O (αm ), there is a total number of (m − 1) factors of δ and W . The δ factors always appear in even-length strings separated by the W factors. Each string of δ of length 2M is associated with a multiplicative coefficient of the M th Catalan number CM . For instance, at O α17 there is a term W δδδδδδW W W δδδδW W , whose prefactor will be C3 C2 = 5 × 2 = 10. We can resum Eq. (C7) by observing that we can uniquely construct every term containing a least one factor of W , by adding to an existing term a (possibly empty) even-length string of δ followed by one factor of W ; e.g. the term δδδδW W δδW is uniquely constructed as δδδδ/W /W δδW . In other words, Π satisfies the relation ¯ Πjj ′ Πjj ′ = + 2 2 l1 l2 ¯ Πl j ′ Πjl1 Wl1 l2 2 . 2 2 (C8) ¯ Here Π is the part of Π which does not contain any factors of W : ¯ αj αj ′ αj Πjj ′ = δjj ′ + 2 2 2 2 = αj δjj ′ 2 ∞ CM M=0 n1 αj 2 αn1 δjn1 δn1 j ′ + 2 n 1 n2 n3 2M αj = 1+ 1 − α2 j αn1 αn2 αn3 2δjn1 δn1 n2 δn2 n3 δn3 j ′ + · · · 2 2 2 δjj ′ . (C9) 23 In the last line we have used the generating function of Catalan numbers,44 ∞ CM xM = M=0 1+ 2 √ . 1 − 4x (C10) Inserting Eq. (C9) into Eq. (C8) and solving for Π, we obtain Eq. (4.3) in the case of spatially uniform interactions, αn (y) = αn . We now argue that the cutoff-dependence of the Luttinger parameter is through Eq. (3.5) as is the case with the first order calculation. To this end, notice that it is values of yn between 0 and O (1/E + ) that dominate the integral ˜ in Eq. (C5). Therefore, when D = Re E + vF n /Ln , the integral is governed by vF n yn ˜ Ln ; in this range of yn , ˜ αn (vF n yn ) = αn (0). On the other hand, when D ≪ vF n /Ln , the integral is controlled mainly by vF n yn ≫ Ln , ˜ ˜ where αn (vF n yn ) = αn (∞). This justifies the crossover behavior given by Eqs. (3.5) and (4.4), and concludes the ˜ calculation of the self-energy terms in the RPA conductance. Calculations of the RPA vertex corrections, or the ring diagrams, are completely in parallel with the first-order vertex corrections except Eq. (C5) appears in the real space integrals. Here E + in Eq. (C5) should be substituted for ω + . At the mth order, all m factors of 1/ω + in Eqs. (C4) and (C6) can be paired with the m + 1 factors of ω + from loop energy integrals; the single unpaired ω + will be combined with the 1/ω factor in Eq. (2.12) so that the conductance is finite in the DC limit. Also, all interaction strengths appearing here are those in the leads αn (∞); this is because in the DC limit ω vF n /Ln for any lead n, and we may refer to our argument in the previous paragraph for D ≪ vF n /Ln . Eventually, taking into account the dressing of the electron lines, we recover Eq. (4.5). Real space integral Eq. (C6) 2. To prove Eq. (C6), we adopt the following change of variables in Eq. (C5): z0 = y1 , z2j−1 = y2j−1 − y2j , z2j = ˜ ˜ ˜ M y2j+1 − y2j , 1 ≤ j ≤ M . The absolute value of the Jacobian of this change of variables is simply (−1) ˜ ˜ j l=0 also introduce the shorthand sj = (−1)l zl . Eq. (C5) then becomes M ∞ IM E + = 1. We = s2l−2 dz0 0 dz2l−1 l=1 M ∞ 0 e2iE dz2l 0 + z2j (C11) j=0 Now consider the auxiliary object, M ˜ IM E + , z0 ≡ s2l−2 l=1 M ∞ dz2l−1 0 e2iE dz2l 0 + z2j j=0 ≡ 2iE + −2M e2iE + M z0 l=0 TM,l −2iE + z0 l! l (C12) + ˜ ˜ where TM,l are dimensionless coefficients; obviously I0 (E + , z0 ) = e2iE z0 and T0,0 = 1. IM obeys the recurrence relation z0 ˜ IM+1 E + , z0 = dz1 e2iE 0 + ∞ z1 0 ˜ dz2 IM E + , z0 − z1 + z2 . Inserting Eq. (C12) into Eq. (C13), we find that TM,l satisfies the simple recurrence relation TM+1,l = and that TM+1,0 = 0 (M ≥ 0). Such a recurrence relation leads to the Catalan’s triangle,44 TM,l = (2M − l − 1)!l (M ≥ 1) . M ! (M − l)! 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